# Leakage Inductance Analytical Calculation for Planar

## Transcript Of Leakage Inductance Analytical Calculation for Planar

Leakage Inductance Analytical Calculation for Planar Components With Leakage Layers

Wenhua Tan, Xavier Margueron, Laurent Taylor, Nadir Idir

To cite this version:

Wenhua Tan, Xavier Margueron, Laurent Taylor, Nadir Idir. Leakage Inductance Analytical Calculation for Planar Components With Leakage Layers. IEEE Transactions on Power Electronics, Institute of Electrical and Electronics Engineers, 2016, 31 (6), pp.4462 - 4473. 10.1109/TPEL.2015.2471090. hal-01886543

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Leakage Inductance Analytical Calculation for Planar Components with Leakage Layers

Wenhua Tan, Xavier Margueron, Member, IEEE, Laurent Taylor, Nadir Idir, Member, IEEE

Abstract—Planar magnetic components are promising solutions for the integration of power electronic systems. The leakage inductance of such components plays an essential role in power converters. In this paper, an analytical modeling method for leakage inductance computation is developped for planar components with plasto-ferrite leakage layers. This method is based on the solution of Poisson’s equations for magneto-static using multilayered Green’s functions. The obtained formulations are general and precise and have been validated by numerical tests. Experimental characterizations have been performed on two magnetic components: A planar LLC and planar common mode choke. The obtained results show that with the described method, the static leakage inductance of planar components can be accurately estimated.

Index Terms—Leakage inductance, planar component, magnetic shunt, Green’s function.

I. INTRODUCTION

T HE trends toward integration of power electronic systems make planar magnetic components prevalent solutions to realize more compact power converters [1], [2]. These components exhibit low proﬁle, high power density and high reproducibility compared to other types of magnetic components. The leakage inductance of such component is a topic of huge interest. Indeed, due to their low proﬁle, their leakage inductance value is usually supposed to be low [3]. For many applications, this low value can be an advantage because transformer’s leakage inductance may cause extra losses, stress on components and ElectroMagnetic Interference (EMI) problems due to high frequency voltage oscillations [4]– [6].

On the opposite, two main applications can ﬁnd the usefulness of a high leakage inductance. For resonant converter LLC (Fig. 1(a)), the leakage inductance Llk can be used for Zero Voltage Switching (ZVS), without adding an extra inductor [7], even if this leakage inductance can also causes voltage imbalance [8] or high voltage oscillations [9]. This leakage inductance must have then a speciﬁc value for typical softswitching converters [10] or Dual Active Bridge (DAB) for example [11]. If its value is not sufﬁcient, some supplementary resonant inductor has to be added [12], [13].

Manuscript received March 13, 2015; revised June 25, 2015; accepted August 8, 2015

W. Tan, X. Margueron and L. Taylor are with Laboratory of Electrical Engineering and Power Electronics, Ecole Centrale de Lille, CS20048, 59650, Villeneuve d’Ascq, France e-mail: [email protected]; [email protected]; [email protected]

N. Idir is with Laboratory of Electrical Engineering and Power Electronics, University of Lille 1, 59655, Villeneuve d’Ascq, France e-mail: [email protected]

For EMI ﬁlters, the leakage inductance of a common-mode (CM) choke can also be used for ﬁltering differential mode (DM) perturbations [14]–[18]. For example, in [14], a leakage layer has been integrated between Printed Circuit Board (PCB) inside a pot core while in [15], the leakage layer has been inserted inside a mixed toroidal/EQ planar core. In both case, this leakage layer enable to increase the leakage inductance of the CM inductor. Other integrated EMI ﬁlters using CM leakage inductance to ﬁlter DM perturbations have also been reported in the past years, including ﬂexible multilayers [16], combined toroidal cores for DM and CM chokes [17] or using magnetic epoxy mixture [18].

As discussed earlier, regarding planar components (transformers or CM chokes), their leakage inductances can be then increased by adding some supplementary leakage layers (also called magnetic shunt) such as Ferrite Polymer Composite (FPC) between the component’s windings [6], [15], [19], [20]. Resonant converters or CM inductors require a speciﬁc amount of leakage inductance so an accurate modeling method for calculating such Llk is necessary.

In the literature, many methods have been reported to assess the leakage inductance of a transformer, based on Finite Element Analysis (FEA) or analytical calculation [21]–[35]. For FEA, computation can be made in 2D, 3D, or mixed 2D/3D for HF losses and inductance values [1], [21]–[23]. FEA is a powerful tool to study electromagnetic components but when dealing with optimization, using such tool can be very cumbersome and time consuming. Even if the numerical resolution is becoming faster with last generation computers, time for geometry’s description and mesh can also be prohibitive. For magnetic component designers, analytical tool will be prefered, in particular for ﬁrst design steps, when transformer parameters have to be tuned as desired. Most of the analytical methods for the Llk calculation of a transformer are based on 1D Dowell’s assumption [24] such that the magnetic ﬂux is tangential to the surface of conductors [25]– [31]. This assumption is true when the magnetic window is well ﬁlled with long conductors. However, in some cases, the winding arrangement may be very irregular such as low ﬁlling factor and non-alignment of conductors. Applying the Dowell’s assumption will lead to errors in the results. To be more general, the method of Roth can be applied for 2D transformer’s winding cross section. This method consists in using double Fourier series to solve the Poisson’s equations in a rectangular region [36]. However, double Fourier series exhibit a low convergence rate so that they are computationally expensive. Besides, this method only works when the matter in the rectangular region is uniform. For planar components

2

(a)

(b)

Fig. 1. Integrated planar LLC. (a) Equivalent circuit. (b) Cross-section of the component.

whose cross-sections are not magnetically homogeneous, like in the case of leakage layers, Roth’s method can not be applied. In such cases, only FEA are performed to determine the value of the leakage inductance [2]. No analytical formulation is available in the literature, except in [37] where some reluctance model is used. This method is still limited to magnetic window well ﬁlled with long conductors.

In [38] and [39], authors propose a general 2D analytical method for calculating the leakage inductance of planar transformers. This method is based on a PEEC-like formula for rectangular conductors. To account for the inﬂuence of magnetic core, the method of magnetic images is employed on the cross-section of the winding part inside the core [40], [41]. This general method, completely independent of the conductor’s arrangement, can accurately evaluate the magnetic ﬁeld in the cross-section of the component for conventional power electronic transformers, planar ones or more recently for power transformers [42]. However, similarly to Roth’s method, the PEEC-like formula also requires a uniform environment of matter as prior condition.

If the cross-sections are magnetically multi-layered, the application of such method can be very cumbersome. For the above reasons, an improved method based on multilayered Green’s function has been developed in this paper for calculating the leakage inductance of planar components having magnetically multi-layered structure.

This paper is organized as follows: in Section II, a review of the leakage inductance calculation and the PEEClike formulations are given. Section III introduces the multilayered Green’s function with detailed analysis and numerical validations. In Section IV, two application examples, LLC and CM choke are presented to validate the proposed method. The paper will end with conclusion.

II. LEAKAGE CALCULATION PROBLEM ANALYSIS

A. Modeling of Leakage Inductance

The existing modeling methods for calculating the static leakage inductance rely on the evaluation of the magnetic energy stored in the component when the total ampere-turns are compensated [43]. The relation between the leakage inductance and the magnetic energy Wmag is expressed by:

Wmag = 1 LlkI2.

(1)

2

Fig. 2. Rectangular conductor cross-section.

where Llk is the static leakage inductance and I the current in the winding. This magnetic energy Wmag (1) can be determined by 2D or 3D ﬁnite element simulations (FEM), which are precise but very time-consuming or with analytical approaches. This computation is generally based on 2D approximation that the current direction is perpendicular to the cross-section of the studied component. Consequently, the magnetic energy Wmag can be calculated by:

Wmag = W smagLmean

(2)

where W smag is the energy per unit length stored in the cross-section of the studied component whereas Lmean is the mean length of the winding ﬁxed from the location of the peak energy density.

In High Frequency (HF), it is well known that the leakage inductance value will decrease due to eddy current effects [34], [35]. In this work, impact of HF magnetic ﬁelds on leakage inductances will not be studied. Indeed, the goal of the developped Green analytic tool is to give designers a fast estimation of their leakage value, as well as in [39]. If this static value does not seem sufﬁcient, the design has to be started again because in HF, leakage will be further reduced.

B. Review of 2D PEEC-like Modeling Method

In 2D magnetostatic analysis, the magnetic ﬁeld over the cross-section of a planar component is governed by the following Poisson’s equation:

∇2Az = −µJz.

(3)

In order to solve this equation, authors proposed in [38], [39] the 2D PEEC-like formulations, where the leakage inductance of planar transformers can be calculated analytically with no hypothesis on the conductors’ topology. In free space, the potential vector Az in the space due to a rectangular conductor carrying current I (Fig. 2) can be expressed by (4).

With this analytical formula, the distribution of the potential vector in the space due to the rectangular conductor can be determined. In case of multiple conductors, the superposition principle can be applied for the calculation. As long as the information of Az is obtained, the integration (5) can be performed to calculate the magnetic energy per unit length stored in the cross-section of the component.

1 W smag = 2 Ω AzJzdxdy (5)

Thus, the leakage inductance of the component can be estimated by:

3

2W smagLmean

Llk ≈

I2

(6)

This method is general and gives accurate estimation of the leakage inductance value. However, just as stated in the introduction, this method can not be applied for multi-layered structure where the permeability of the window is not uniform. Although the magnetic image method [40], [41] can applied for taking into account the inﬂuence of magnetic material, it can not be applied on a multi-layered structure where a ferrite polymer layer is present, for the following reasons:

1) With this method, four ﬁrst-order images are employed due to direct reﬂections and four second-order images are added due to secondary reﬂections [39]. Higher order images are neglected. However, when FPC layer is implemented in the region, the reﬂection phenomenon becomes much more complicated.

2) The aforementioned method assumes that the thickness of the ferrite is inﬁnite. As long as the permeability of the ferrite material is large enough, this assumption is very close to the reality. However, since the FPC layer has a low permeability, the impact of its thickness has to be considered, which further complicates the calculation.

In view of these reasons, an analytical method for leakage inductance calculation based on Green’s function for multilayered structure is introduced.

III. MULTI-LAYERED GREEN’S FUNCTION

A. Description of the Problem

The structure of planar components with PCB technology is shown in Fig. 1(b). The leakage layer in ferrite polymer composite is optional but can be used to increase the leakage inductance for passive component integration in the application of soft-switching converters or integrated CM and DM inductors for EMI ﬁlters. It should be noted that the FR-4 epoxy and the isolating layers are considered as magnetically transparent. As a consequence, the component cross-section is composed of three layers of matter, i.e. transparent-FPCtransparent. Similarly to [38], the current is supposed to be perpendicular to this cross-section and the system is described by the magnetostatic Poisson’s equation (3). As the relative permeability of MnZn ferrite material is very high (in order of 103 − 104), it can be considered as magnetic material with inﬁnite permeability as an approximation. It can be shown that the boundary conditions (8) holds by using the continuity conditions on the tangential component of the magnetic ﬁeld H (Fig. 3):

HfTerrite − HwTindow = 0

(7)

Fig. 3. Continuity conditions on the boundary in the core window.

The boundary conditions for the problem are derived as follows:

∂Az

= 0 over the horizontal surface

∂∂Ayz (8)

= 0 over the vertical surface

∂x

As seen, homogeneous Neumann boundary conditions are obtained. However, this problem admits solutions if and only if the compatibility condition is satisﬁed:

∂Az dl + µ0 JzdS = 0 (9)

∂Ω ∂n

Ω

This can be proved by applying Gauss’s Law on the Poisson’s equation (3). With the homogeneous boundary conditions (8), the ﬁrst term of equation (9) is null, requiring the second term to be null too. In fact, this is the prior condition for the calculation of leakage inductance. Therefore, the Poisson’s equation (3) with these boundary conditions admit solutions.

After solving the potential vector Az from (9), the equation (5) and (6) should be used to determine the leakage inductance of such component.

B. Generality on Green’s Function

Green’s function is a powerful tool for analyzing electromagnetic problems. It is a specially constructed function to solve partial differential equations [44]. It can be regarded as the impulse response of an electromagnetic system to Dirac type excitation. For the magnetostatic Poisson’s equation given in (3), its associated Green’s function is:

∇2G = −µδ

(10)

where δ is a Dirac current source. With this Green’s function, the solution of the original problem can be expressed by:

1

∂G

Az (x, y) = Ω G (x, y) JzdS − µ ∂Ω Az ∂n dl (11)

X=x−a Y =y−b

Az (x, y; a, b) = −µ0I XY ln X2 + Y 2 + X2 arctan Y + Y 2 arctan X

(4)

16πab

X Y X=x+a Y =y+b

4

From this equation, it is desirable to also provide homogeneous Neumann boundary conditions for the Green’s function so that the second term of (11) is null and the potential vector Az can be represented by a convolution. However, this is not allowed since such boundary condition will violate the compatibility condition. Indeed, the following compatibility condition should be satisﬁed during the construction of the Green’s function:

∂G ∂Ω ∂n dl = −µ0 (12)

C. Multi-layered Green’s Function

In [45], [46], authors present a multi-layered Green’s function for solving electrostatic Poisson’s equation with homogeneous Dirichlet boundary conditions in a bounded multilayered region. As stated previously, the studied planar components have a magnetically multi-layered cross-section. In this paper, the method of [45] will be extended for magnetostatic analysis. It has been shown that the boundary conditions should verify (8) to admit solutions. Therefore, the following boundary conditions are proposed:

∂G

µ0

=−

∂y y=H

L

(13)

∂G

∂G

∂G

=

=

=0

∂x x=0 ∂x x=L ∂y y=0

As seen, only the top edge of the rectangular region is not zero. With these boundary conditions, the compatibility conditions given in (12) can be satisﬁed.

According to the previous analysis, the cross-section of the planar component is a 3-layer structure. If a Dirac current source is introduced, the structure is further split into four layers. Therefore, only a 4-layer structure is studied in this work. As shown in Fig. 4(a), each layer has a height Hk and a permeability µk. Here, the local coordinates are applied for each layer and the origin is chosen at the lower left vertex of each rectangular region. The Dirac current source locates at the interface between layer j and j + 1 (j < 4). As a consequence, the Green’s function in layer i satisﬁes:

∇2Gi = 0

(14)

The following boundary conditions can be written out:

1) Left and right boundaries of layer i: x = 0 and x = L:

∂Gi

∂Gi

=

=0

(15)

∂x x=0 ∂x x=L

2) Top and bottom boundaries (in layer 4 and layer 1):

∂G1

=0

Layer 1

∂y y=0

(16)

∂G4

µ0

= − Layer 4

∂y y=H

L

4

3) Continuity conditions on the interface between layer i

(a)

(b)

Fig. 4. 4-layer structure for Green’s function deduction. (a) Source point higher than FPC layer. (b) Source point lower than FPC Layer.

and i + 1 (i < 4):

Gi

− Gi+1 = 0

y=Hi

y=0

1 ∂Gi

1 ∂Gi+1

−

= δijδ (x − xs)

µi ∂y y=H µi+1 ∂y y=0

i

(17)

where δij is the Chronecker Delta function whereas δ(x − xs) presents the Dirac current source.

From these boundary conditions (15), the general solution in layer i can be derived as follows:

∞

Gi = α0i y + β0i +

αni ch (kny) + βni sh (kny) cos (knx)

n=1

(18)

where ch() and sh() are hyperbolic cosine and sinus functions respectively. Note that the term α0i y + β0i appears since the eigenvalue of (14) can be zero [46].

In order to determine the parameters α0i , β0i , αni and βni , the remaining boundary conditions are used. From (16), the

following relations hold:

α01 = 0 and βn1 = 0

α4 = − µ0 and β4 = −α4 tanh (knH4) (19)

0

L

n

n

The next step consists in using the continuity condition

(17) to establish the link between the parameters of different

layers. The detailed derivation is given in Appendix, only the ﬁnal results are given here. The parameters α0i , β0i , αni and βni can be expressed by (24). Note that Fln,l+1 and Fln,l+1 −1 are the up-going and down-going transformation matrix, respectively:

Fl0,l+1 =

0 µl+1

µl

Hl 1

(20)

−1

Fl0,l+1 =

µl

0 µl+1

−µl

H 1 µl+1 l

(21)

Fln,l+1 =

ch (knHl) µµl+l 1 sh (knHl)

sh (knHl) µµl+l 1 ch (knHl)

(22)

5

l,l+1 −1

ch (knHl) − µµl sh (knHl)

Fn = −sh (knHl) µµl+ll+11ch (knHl) (23)

The parameter αn1 and αn4 can be derived from the continuity condition when i = j.

D. Magnetic Field Calculation

The ﬁrst integral can be performed on the conductor area where the current is non-zero. The second term corresponds to the average value of the potential vector on the boundary of the region, which is a constant. Neglecting this constant, the ﬁrst integral of (11) can be evaluated analytically [due to the term cos(knxs) in (25)] along x-axis and numerically along y-axis by discretizing the conductor into K thin elements (Fig.5). For element k, the y coordinate of the center point ysm = (yk + yk+1)/2 is chosen for the integration along y-axis.

The magnetic induction B can then be calculated by deriving Az, as given by (26). Note that the parameters α0i , β0i , αni and βni are only dependent to the coordinate of the source point. This property allows the separation of the mathematical treatment on the source conductor and the observation one. Therefore, the partial derivation on G (on the observation point) given in (26) can be performed directly on the expression whereas the integration on the source conductor is still evaluated numerically as done before.

∂Az

∂G (x, y)

Bx =

= −µ

J ds

∂y

Ω ∂y

(26)

∂Az

∂G (x, y)

By = − = µ

J ds

∂x

Ω ∂x

The ﬁnal purpose involves estimating the total energy in the cross-section via the integral (5). The Greens function has to be integrated both on the source conductor and the observation conductor. The integration on the source conductor is performed numerically while the integration on the observation conductor can be performed analytically according to the expression (18). However, three cases should be distinguished (Fig. 6):

1) The position of the source conductor element is higher than the observation conductor: In this case, the observation conductor is completely in layer i+1, therefore the integration is performed with the Greens function Gi+1 for layer i+1.

2) The position of the source conductor element is lower than the observation conductor: In this case, the observation conductor is completely in layer i, the integration is thereby calculated with the Greens function Gi for layer i.

3) The position of the source conductor element is included in the observation conductor: In this case, as the observation conductor crosses the two layers, the integration is performed by two separate parts. Both the Greens function Gi and Gi+1 should be integrated on the corresponding parts.

Fig. 5. Discretization along y-axis of the source conductor.

Fig. 6. Analytical integration on the observation conductor: Three cases.

Note that during the calculation of Az, a constant appears on the ﬁnal obtained results due to the Neumann type boundary conditions [47]. However, this constant will not affect the results of (5) since the integral of this constant over all the conductors will be null as the total ampere-turns is compensated in the component’s window.

E. Numerical Validations To validate the proposed method, the multi-layered Green’s

function is applied on two structures. The ﬁrst case (Fig. 7) does not correspond to real component but is computed to test the developped approach. The second one, is closer to a real planar component (Fig. 10(a)). For both components, numerical simulations, performed with ANSYS Maxwell 2D [48], are done to be compared to Green’s calculation. For each FEA simulation, total ampere-turns are compensated.

1) Conﬁguration test: As shown in Fig. 7, an FPC layer is implemented so that the structure is divided into three layers. The total ampere-turns of the component are compensated. To analyse the results, two test cuts are performed on this crosssection at y = 0.75 mm and y = 3.75 mm respectively. The obtained vector potential on these two test cuts are compared with Finite Elements simulations results, as shown in Fig. 8(a). As seen, a constant difference is observed between the calculated curves and the simulated ones. These results

6

Fig. 7. Test conﬁguration.

(a)

(b)

Fig. 8. Comparison between calculated results and simulated results. (a) Before removing the constant. (b) After removing the constant.

are foreseen due to the Neumann type boundary conditions. Removing this constant (Fig. 8(b)), a good agreement between these two curves can be found. The magnetic induction Bx and By can also be deduced, as illustrated in Fig. 9. One can note a good coherence between the calculated curves and simulated ones. After the numerical integration, the magnetic energy stored in the window is 2.29E − 5J/m while the calculated one is similar. In the previous analysis, it is supposed that the permeability is inﬁnite. Indeed, even though the practical permeability of ferrite material is in the order of 103µ0, the inﬁnite-permeability assumption still gives good approximation to the reality. FEA simulations have been performed to assess the inﬂuence of the magnetic core. Results are presented in Table I where the magnetic energy stored in the windows is computed with various permeability values. Comparing FEA results and the Green’s calculations, it can be seen that the proposed method can accurately determine the magnetic ﬁeld in the structures of interest.

2) Planar CM choke: This second test case is based on the design of a planar CM choke (Fig. 10(a)) made of 3F3 E32/I planar core [49]. Some FPC layer (height 0.8mm) is added between two windings. The total ampere-turns of the component are also compensated. The current in each conductor higher than the FPC layer is −1A whereas the current is 1A for the conductors lower than the FPC layer. The Fig 10(b) shows the magnetic ﬁeld inside the component’s window. From this simulation, it can be concluded that the ﬁeld is not tangential to the conductors. Therefore, the 1D

Dowell assumption is not directly applicable in such case. This hypothesis is conﬁrmed when analysing Bx and By (Fig. 11) along the horizontal test cut at H = 1.5mm. Simulated results are also compared to the computed ones in Fig. 11, where both simulated and calculated ﬁelds show good agreement.

After numerical integration, the calculated magnetic energy stored in the window is 2.43E − 5J/m, while the simulated value is 2.57E − 5J/m. The difference is equal only to 5%.

IV. APPLICATIONS

The previous method has been tested on two real components: A LLC transformer and a CM choke. Both devices are based on the use of ferrite planar cores [49], associated with FPC C350 [50] for the leakage layers.

A. Components’ description

For the LLC transformer (Fig. 12(a)), two 3F3 E38-cores are used while for the CM inductor (Fig. 13(a)), two 3C90 E43-cores are preferred.

The transformer is made of 4 copper layers of 35µm with 8 turns for the primary and 2 for the secondary. The CM inductor is a symmetrical component, made of 8 identical 70µm copper layers with 4 turns on each layer. For both components a leakage layer made of FPC is sandwiched between the two windings to increase the leakage inductance. This FPC layer is set to 0.2mm and 0.96mm, for the transformer and the CM inductor, respectively. The winding arrangements are described in Fig.12(b) and Fig.13(c).

0

α0i

=

0

β0i N−1 Fl0,l+1 −1

l=i

− µL0 0

i≤j and

i>j

1

Fl,l+1

αn1

αni

=

l=i−1

n

0

βni N−1

−1

α4

Fln,l+1

n

−α4 tanh (k H )

l=i

n

n4

j

−1

Fl0,l+1

l=3

j

Fl,l+1 −1

n

l=3

− µL0 β04

j

=

Fl0,l+1

l=1

0

− µj+1

β1 +

L

0

0

αn4 −αn4 tanh (knH4)

j

=

Fln,l+1

l=1

αn1 + 0

0 − 2µnjπ+1 cos (knxs)

i≤j i>j

(24)

(25)

7

TABLE I VARIATION OF MAGNETIC ENERGY WITH PERMEABILITY

µr

∞

10000 5000 1000

500

W smag(J/m) 2.29E-5 2.29E-5 2.29E-5 2.28E-5 2.27E-5

(a)

(a)

(b)

(c)

(d)

Fig. 9. Comparison between calculated results and simulated results for the test case. (a) Test cut 1: Bx. (b) Test cut 1: By. (c) Test cut 2: Bx. (d) Test cut 2: By.

(b)

Fig. 10. Planar CM choke cross section under simulation. (a) Window cross section. (b) Field inside the window.

B. Leakage inductance calculation

For the part inside the core, the previous Green’s computation is applied. For the winding part outside the core, the FPC layer is not implemented. Therefore, the PEEC-like formulations can be employed for calculating the magnetic ﬁeld as well as the magnetic energy per unit length. In fact, it can be shown that the PEEC-like formula (4) is the integral form of the free space Green’s function (Green’s theorem) over the rectangular region [−a/2, a/2] × [−b/2, b/2] [51]:

G (x, y; xs, ys) = − µ0 ln (x − xs)2 + (y − ys)2 (27) 2π

The Table II presents energy computed inside the core using Green’s functions Wic and the one calculated outside the core based on PEEC formulas Woc. The total leakage inductance is then calculated with (28) taking into account lengths of windings inside and outside the core (Fig. 14). For the LLC transformer, the leakage inductance is evaluated to 162nH while for the CM inductor, it is equal to 13.4µH.

2

Llk = I2 [2Wiclic + Woc(loc1 + loc2)]

(28)

Regarding computation time, duration depends on the num-

ber of conductors. For example, the LLC computation lasts 3.45s while the CM inductor 5.14s. As a comparison, FEA performed on the example of Fig. 10(a) lasts 15s. Such semi-

(a)

(b)

Fig. 11. Comparison between calculated results and simulated results. (a) Test cut: Bx. (b) Test cut: By.

analytical approach can then be interesting for component’s optimization and its leakage inductance tuning.

C. Leakage inductance measurement

Both components are then characterized with an impedance analyzer [52] and their leakage inductances are measured based on method proposed in [53]: The leakage impedance Zs (Fig. 15) is calculated using open-circuit and short-circuit impedance measurements (29). This method enables to cancel the effect on magnetizing inductance while calculating the leakage one (30).

Zs = Z0 (Zcc − rp) (29) (Z0 − rp) − rs

8

(a)

(b)

Fig. 12. LLC planar transformer (a) Component top view. (b) Cross section and dimensions.

Fig. 14. Typical winding length.

Fig. 15. Equivalent circuit for leakage measurement.

(a)

(b)

(c)

Fig. 13. Planar CM choke. (a) Component Top view. (b) Component Face view. (c) Cross section and dimensions

with Z0 the impedance measured from primary winding with secondary in open-circuit, Z0 the impedance measured from secondary winding with primary in open-circuit and Zcc the impedance measured from primary winding with secondary in short-circuit. rp (Fig.15) is deduced from Z0 in low frequency while rs is calculated with Z0 also in low frequency [53].

TABLE II LEAKAGE INDUCTANCE CALCULATION

Component Core type Mag. material

lic (m) loc1 + loc2 (m)

Wic (J/m) Woc (J/m)

Llk (H)

LLC E38 3F3 25.4E − 3 30E − 3 1.009E − 6 5.007E − 7 162E − 9

CM ind E43 3C90

27.9E − 3 88E − 3 6.691E − 5 3.367E − 5 13.4E − 6

Zs

Llkmeas

=

I m( 2πf

)

(30)

The measured values are plotted in Fig. 16, where they are compared to the computed ones. In this ﬁgure, for low frequencies, the leakage inductance is masked by resistances while in HF, parasitic capacitances cancel the magnetic effect. The leakage value is quite constant on the frequency range [1kHz; 1M Hz].

In both components coupling coefﬁcients k between windings are close to 1. Magnetizing inductances are evaluated to 431µH and 851µH for the LLC and the CM inductor, respectively. These values are widely stronger than the leakage ones. With such a difference, the total ampere-turn compensation hypothesis stays valid because magnetizing current can be neglected in case of short-circuit test.

D. Discussion

Regarding Fig. 16, it can be noted that computation for both cases gives good result with error lower than 15%. These differences can be justiﬁed for following reasons:

1) Errors related with dimensions: The dimensions reported in Fig. 12(b) and Fig. 13(c) have been measured directly on the prototypes and are not precises.

2) Simpliﬁcation of the geometry: The window’s description is not identical inside and outside the core. The part inside the magnetic core is well described but the one outside is simpliﬁed and the angles/corners are not really taken into account.

3) Sensitivity to the average length: The results are directly proportional to the energy density. They have to be multiplied by a lenght which is easy to determine inside the core (depth of the core) but which is more difﬁcult

9

to evaluate outside. A mean path is ﬁxed, based on window’s peak energy but this assumption is not precise. 4) Magnetic material disparities: As an example, the C350 material is given with a permeability of 9 ± 20% [50]. The same computation has been done again, taking into account this disparity. Calculation has been done with µr = 10.8(9 + 20%) and µr = 7.2(9 − 20%). The Table III summarizes the results and shows that this parameter can lead to signiﬁcant differences.

Without the leakage layers, leakage inductance can be evaluated, based on [39], to 128nH (Wic = 6.68E − 7J/m) and 11.8µH (Wic = 5.287E−5J/m) for the LLC transformer and the CM inductor, respectively. The leakage layers enable then to increase the leakage inductance and can be analytically calculated with a good precision with the developped method. However, this increase can be considered as quite low while regarding relative differences between leakage inductances with and without leakage layers. As a percentage, this rise is equal to 26% for the LLC and 13% for the CM inductor. These values can still be increased by acting on two main factors:

1) Increase of FPC thickness: In both examples, FPC layers are limited to 0.2mm and 0.96mm. Adding some layers, can enable to increase the leakage inductance value.

2) Addition of FPC outside the core: In this work, only FPC inside the core was considered. Adding leakage layers outside the core will also increase the leakage value. But the present method will not be able to calculate its value. Only 3D FEA could be performed to do this assessment.

These both solutions can also be discussed. Indeed, in most of power electronic converters, while power density is required, HF power transformers are made of interleaved layers to limit HF copper losses [54], [55]. Such interleaving reduces the leakage inductance. FPC can be added between each primary and secondary layers [56] but here also, the present method will be unusable. Some developpements have to be made on massively multi-layered windings as well as HF effects.

In connection with HF power transformers, problem of parallel windings should also be adressed. In planar transformers, parallel layers are a common practice to increase currents inside windings. The Green method can be applied with such conductor conﬁgurations. Indeed, only current values have to be settled, as in the example presented in Fig.7. In low frequency, the current repartition will be homogeneous and current will be uniformly distributed inside conductors. It is not true in HF, but the developped Green method is also limited and can not be applied in HF.

V. CONCLUSION

In this paper, the leakage inductance of planar magnetic components including leakage layer is studied via an analytical method based on multilayered Green’s function. Using this Green’s function, the Poisson equation for a rectangular region with Neumanns type boundary conditions can be solved to determine the magnetic ﬁeld in the cross-section of the studied component. Simulation and measurement results show that this

Fig. 16. Comparison between measurement and computation results.

method is very effective and accurate for planar component leakage inductance calculation when a leakage layer is inserted inside the component’s windings.

APPENDIX A DERIVATION OF EQUATION

Recall the general solution of Greens function in layer i (18) and the continuity condition (17). If i = j, δij = 0. In this case, the Dirac current source does not locate on the interface to analyze. The condition can be expressed by (31). Based on the equality of the coefﬁcients term by term, the following matrix systems can be derived:

αni+1 = Fi,i+1 αni

βni+1

n

βni

(n = 0, 1, 2...∞) (33)

Here the matrix Fin,i+1 is the upward transformation matrix that relates the parameters of layer i and layer i+1.

Its expression can be found in (20) and (22). The downward transformation matrix Fin,i+1 −1 can be deﬁned similarly as:

αni = Fi,i+1 −1 αni+1

βni

n

βni+1

(n = 0, 1, 2...∞) (34)

where Fin,i+1 −1 is given in (21) and (23). If i = j, δij = 1. In this case, the Dirac current source locates on the studied interface. The condition (17) becomes (32). Integrate the second term of this equation (32) on [0, L]:

L αj − L αj+1 = 1

(35)

µj 0 µj+1 0

Multiply the two sides of the second equation of (32) by cos(knx), and integrate it on [0, L]:

Wenhua Tan, Xavier Margueron, Laurent Taylor, Nadir Idir

To cite this version:

Wenhua Tan, Xavier Margueron, Laurent Taylor, Nadir Idir. Leakage Inductance Analytical Calculation for Planar Components With Leakage Layers. IEEE Transactions on Power Electronics, Institute of Electrical and Electronics Engineers, 2016, 31 (6), pp.4462 - 4473. 10.1109/TPEL.2015.2471090. hal-01886543

HAL Id: hal-01886543 https://hal.archives-ouvertes.fr/hal-01886543

Submitted on 2 Oct 2018

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1

Leakage Inductance Analytical Calculation for Planar Components with Leakage Layers

Wenhua Tan, Xavier Margueron, Member, IEEE, Laurent Taylor, Nadir Idir, Member, IEEE

Abstract—Planar magnetic components are promising solutions for the integration of power electronic systems. The leakage inductance of such components plays an essential role in power converters. In this paper, an analytical modeling method for leakage inductance computation is developped for planar components with plasto-ferrite leakage layers. This method is based on the solution of Poisson’s equations for magneto-static using multilayered Green’s functions. The obtained formulations are general and precise and have been validated by numerical tests. Experimental characterizations have been performed on two magnetic components: A planar LLC and planar common mode choke. The obtained results show that with the described method, the static leakage inductance of planar components can be accurately estimated.

Index Terms—Leakage inductance, planar component, magnetic shunt, Green’s function.

I. INTRODUCTION

T HE trends toward integration of power electronic systems make planar magnetic components prevalent solutions to realize more compact power converters [1], [2]. These components exhibit low proﬁle, high power density and high reproducibility compared to other types of magnetic components. The leakage inductance of such component is a topic of huge interest. Indeed, due to their low proﬁle, their leakage inductance value is usually supposed to be low [3]. For many applications, this low value can be an advantage because transformer’s leakage inductance may cause extra losses, stress on components and ElectroMagnetic Interference (EMI) problems due to high frequency voltage oscillations [4]– [6].

On the opposite, two main applications can ﬁnd the usefulness of a high leakage inductance. For resonant converter LLC (Fig. 1(a)), the leakage inductance Llk can be used for Zero Voltage Switching (ZVS), without adding an extra inductor [7], even if this leakage inductance can also causes voltage imbalance [8] or high voltage oscillations [9]. This leakage inductance must have then a speciﬁc value for typical softswitching converters [10] or Dual Active Bridge (DAB) for example [11]. If its value is not sufﬁcient, some supplementary resonant inductor has to be added [12], [13].

Manuscript received March 13, 2015; revised June 25, 2015; accepted August 8, 2015

W. Tan, X. Margueron and L. Taylor are with Laboratory of Electrical Engineering and Power Electronics, Ecole Centrale de Lille, CS20048, 59650, Villeneuve d’Ascq, France e-mail: [email protected]; [email protected]; [email protected]

N. Idir is with Laboratory of Electrical Engineering and Power Electronics, University of Lille 1, 59655, Villeneuve d’Ascq, France e-mail: [email protected]

For EMI ﬁlters, the leakage inductance of a common-mode (CM) choke can also be used for ﬁltering differential mode (DM) perturbations [14]–[18]. For example, in [14], a leakage layer has been integrated between Printed Circuit Board (PCB) inside a pot core while in [15], the leakage layer has been inserted inside a mixed toroidal/EQ planar core. In both case, this leakage layer enable to increase the leakage inductance of the CM inductor. Other integrated EMI ﬁlters using CM leakage inductance to ﬁlter DM perturbations have also been reported in the past years, including ﬂexible multilayers [16], combined toroidal cores for DM and CM chokes [17] or using magnetic epoxy mixture [18].

As discussed earlier, regarding planar components (transformers or CM chokes), their leakage inductances can be then increased by adding some supplementary leakage layers (also called magnetic shunt) such as Ferrite Polymer Composite (FPC) between the component’s windings [6], [15], [19], [20]. Resonant converters or CM inductors require a speciﬁc amount of leakage inductance so an accurate modeling method for calculating such Llk is necessary.

In the literature, many methods have been reported to assess the leakage inductance of a transformer, based on Finite Element Analysis (FEA) or analytical calculation [21]–[35]. For FEA, computation can be made in 2D, 3D, or mixed 2D/3D for HF losses and inductance values [1], [21]–[23]. FEA is a powerful tool to study electromagnetic components but when dealing with optimization, using such tool can be very cumbersome and time consuming. Even if the numerical resolution is becoming faster with last generation computers, time for geometry’s description and mesh can also be prohibitive. For magnetic component designers, analytical tool will be prefered, in particular for ﬁrst design steps, when transformer parameters have to be tuned as desired. Most of the analytical methods for the Llk calculation of a transformer are based on 1D Dowell’s assumption [24] such that the magnetic ﬂux is tangential to the surface of conductors [25]– [31]. This assumption is true when the magnetic window is well ﬁlled with long conductors. However, in some cases, the winding arrangement may be very irregular such as low ﬁlling factor and non-alignment of conductors. Applying the Dowell’s assumption will lead to errors in the results. To be more general, the method of Roth can be applied for 2D transformer’s winding cross section. This method consists in using double Fourier series to solve the Poisson’s equations in a rectangular region [36]. However, double Fourier series exhibit a low convergence rate so that they are computationally expensive. Besides, this method only works when the matter in the rectangular region is uniform. For planar components

2

(a)

(b)

Fig. 1. Integrated planar LLC. (a) Equivalent circuit. (b) Cross-section of the component.

whose cross-sections are not magnetically homogeneous, like in the case of leakage layers, Roth’s method can not be applied. In such cases, only FEA are performed to determine the value of the leakage inductance [2]. No analytical formulation is available in the literature, except in [37] where some reluctance model is used. This method is still limited to magnetic window well ﬁlled with long conductors.

In [38] and [39], authors propose a general 2D analytical method for calculating the leakage inductance of planar transformers. This method is based on a PEEC-like formula for rectangular conductors. To account for the inﬂuence of magnetic core, the method of magnetic images is employed on the cross-section of the winding part inside the core [40], [41]. This general method, completely independent of the conductor’s arrangement, can accurately evaluate the magnetic ﬁeld in the cross-section of the component for conventional power electronic transformers, planar ones or more recently for power transformers [42]. However, similarly to Roth’s method, the PEEC-like formula also requires a uniform environment of matter as prior condition.

If the cross-sections are magnetically multi-layered, the application of such method can be very cumbersome. For the above reasons, an improved method based on multilayered Green’s function has been developed in this paper for calculating the leakage inductance of planar components having magnetically multi-layered structure.

This paper is organized as follows: in Section II, a review of the leakage inductance calculation and the PEEClike formulations are given. Section III introduces the multilayered Green’s function with detailed analysis and numerical validations. In Section IV, two application examples, LLC and CM choke are presented to validate the proposed method. The paper will end with conclusion.

II. LEAKAGE CALCULATION PROBLEM ANALYSIS

A. Modeling of Leakage Inductance

The existing modeling methods for calculating the static leakage inductance rely on the evaluation of the magnetic energy stored in the component when the total ampere-turns are compensated [43]. The relation between the leakage inductance and the magnetic energy Wmag is expressed by:

Wmag = 1 LlkI2.

(1)

2

Fig. 2. Rectangular conductor cross-section.

where Llk is the static leakage inductance and I the current in the winding. This magnetic energy Wmag (1) can be determined by 2D or 3D ﬁnite element simulations (FEM), which are precise but very time-consuming or with analytical approaches. This computation is generally based on 2D approximation that the current direction is perpendicular to the cross-section of the studied component. Consequently, the magnetic energy Wmag can be calculated by:

Wmag = W smagLmean

(2)

where W smag is the energy per unit length stored in the cross-section of the studied component whereas Lmean is the mean length of the winding ﬁxed from the location of the peak energy density.

In High Frequency (HF), it is well known that the leakage inductance value will decrease due to eddy current effects [34], [35]. In this work, impact of HF magnetic ﬁelds on leakage inductances will not be studied. Indeed, the goal of the developped Green analytic tool is to give designers a fast estimation of their leakage value, as well as in [39]. If this static value does not seem sufﬁcient, the design has to be started again because in HF, leakage will be further reduced.

B. Review of 2D PEEC-like Modeling Method

In 2D magnetostatic analysis, the magnetic ﬁeld over the cross-section of a planar component is governed by the following Poisson’s equation:

∇2Az = −µJz.

(3)

In order to solve this equation, authors proposed in [38], [39] the 2D PEEC-like formulations, where the leakage inductance of planar transformers can be calculated analytically with no hypothesis on the conductors’ topology. In free space, the potential vector Az in the space due to a rectangular conductor carrying current I (Fig. 2) can be expressed by (4).

With this analytical formula, the distribution of the potential vector in the space due to the rectangular conductor can be determined. In case of multiple conductors, the superposition principle can be applied for the calculation. As long as the information of Az is obtained, the integration (5) can be performed to calculate the magnetic energy per unit length stored in the cross-section of the component.

1 W smag = 2 Ω AzJzdxdy (5)

Thus, the leakage inductance of the component can be estimated by:

3

2W smagLmean

Llk ≈

I2

(6)

This method is general and gives accurate estimation of the leakage inductance value. However, just as stated in the introduction, this method can not be applied for multi-layered structure where the permeability of the window is not uniform. Although the magnetic image method [40], [41] can applied for taking into account the inﬂuence of magnetic material, it can not be applied on a multi-layered structure where a ferrite polymer layer is present, for the following reasons:

1) With this method, four ﬁrst-order images are employed due to direct reﬂections and four second-order images are added due to secondary reﬂections [39]. Higher order images are neglected. However, when FPC layer is implemented in the region, the reﬂection phenomenon becomes much more complicated.

2) The aforementioned method assumes that the thickness of the ferrite is inﬁnite. As long as the permeability of the ferrite material is large enough, this assumption is very close to the reality. However, since the FPC layer has a low permeability, the impact of its thickness has to be considered, which further complicates the calculation.

In view of these reasons, an analytical method for leakage inductance calculation based on Green’s function for multilayered structure is introduced.

III. MULTI-LAYERED GREEN’S FUNCTION

A. Description of the Problem

The structure of planar components with PCB technology is shown in Fig. 1(b). The leakage layer in ferrite polymer composite is optional but can be used to increase the leakage inductance for passive component integration in the application of soft-switching converters or integrated CM and DM inductors for EMI ﬁlters. It should be noted that the FR-4 epoxy and the isolating layers are considered as magnetically transparent. As a consequence, the component cross-section is composed of three layers of matter, i.e. transparent-FPCtransparent. Similarly to [38], the current is supposed to be perpendicular to this cross-section and the system is described by the magnetostatic Poisson’s equation (3). As the relative permeability of MnZn ferrite material is very high (in order of 103 − 104), it can be considered as magnetic material with inﬁnite permeability as an approximation. It can be shown that the boundary conditions (8) holds by using the continuity conditions on the tangential component of the magnetic ﬁeld H (Fig. 3):

HfTerrite − HwTindow = 0

(7)

Fig. 3. Continuity conditions on the boundary in the core window.

The boundary conditions for the problem are derived as follows:

∂Az

= 0 over the horizontal surface

∂∂Ayz (8)

= 0 over the vertical surface

∂x

As seen, homogeneous Neumann boundary conditions are obtained. However, this problem admits solutions if and only if the compatibility condition is satisﬁed:

∂Az dl + µ0 JzdS = 0 (9)

∂Ω ∂n

Ω

This can be proved by applying Gauss’s Law on the Poisson’s equation (3). With the homogeneous boundary conditions (8), the ﬁrst term of equation (9) is null, requiring the second term to be null too. In fact, this is the prior condition for the calculation of leakage inductance. Therefore, the Poisson’s equation (3) with these boundary conditions admit solutions.

After solving the potential vector Az from (9), the equation (5) and (6) should be used to determine the leakage inductance of such component.

B. Generality on Green’s Function

Green’s function is a powerful tool for analyzing electromagnetic problems. It is a specially constructed function to solve partial differential equations [44]. It can be regarded as the impulse response of an electromagnetic system to Dirac type excitation. For the magnetostatic Poisson’s equation given in (3), its associated Green’s function is:

∇2G = −µδ

(10)

where δ is a Dirac current source. With this Green’s function, the solution of the original problem can be expressed by:

1

∂G

Az (x, y) = Ω G (x, y) JzdS − µ ∂Ω Az ∂n dl (11)

X=x−a Y =y−b

Az (x, y; a, b) = −µ0I XY ln X2 + Y 2 + X2 arctan Y + Y 2 arctan X

(4)

16πab

X Y X=x+a Y =y+b

4

From this equation, it is desirable to also provide homogeneous Neumann boundary conditions for the Green’s function so that the second term of (11) is null and the potential vector Az can be represented by a convolution. However, this is not allowed since such boundary condition will violate the compatibility condition. Indeed, the following compatibility condition should be satisﬁed during the construction of the Green’s function:

∂G ∂Ω ∂n dl = −µ0 (12)

C. Multi-layered Green’s Function

In [45], [46], authors present a multi-layered Green’s function for solving electrostatic Poisson’s equation with homogeneous Dirichlet boundary conditions in a bounded multilayered region. As stated previously, the studied planar components have a magnetically multi-layered cross-section. In this paper, the method of [45] will be extended for magnetostatic analysis. It has been shown that the boundary conditions should verify (8) to admit solutions. Therefore, the following boundary conditions are proposed:

∂G

µ0

=−

∂y y=H

L

(13)

∂G

∂G

∂G

=

=

=0

∂x x=0 ∂x x=L ∂y y=0

As seen, only the top edge of the rectangular region is not zero. With these boundary conditions, the compatibility conditions given in (12) can be satisﬁed.

According to the previous analysis, the cross-section of the planar component is a 3-layer structure. If a Dirac current source is introduced, the structure is further split into four layers. Therefore, only a 4-layer structure is studied in this work. As shown in Fig. 4(a), each layer has a height Hk and a permeability µk. Here, the local coordinates are applied for each layer and the origin is chosen at the lower left vertex of each rectangular region. The Dirac current source locates at the interface between layer j and j + 1 (j < 4). As a consequence, the Green’s function in layer i satisﬁes:

∇2Gi = 0

(14)

The following boundary conditions can be written out:

1) Left and right boundaries of layer i: x = 0 and x = L:

∂Gi

∂Gi

=

=0

(15)

∂x x=0 ∂x x=L

2) Top and bottom boundaries (in layer 4 and layer 1):

∂G1

=0

Layer 1

∂y y=0

(16)

∂G4

µ0

= − Layer 4

∂y y=H

L

4

3) Continuity conditions on the interface between layer i

(a)

(b)

Fig. 4. 4-layer structure for Green’s function deduction. (a) Source point higher than FPC layer. (b) Source point lower than FPC Layer.

and i + 1 (i < 4):

Gi

− Gi+1 = 0

y=Hi

y=0

1 ∂Gi

1 ∂Gi+1

−

= δijδ (x − xs)

µi ∂y y=H µi+1 ∂y y=0

i

(17)

where δij is the Chronecker Delta function whereas δ(x − xs) presents the Dirac current source.

From these boundary conditions (15), the general solution in layer i can be derived as follows:

∞

Gi = α0i y + β0i +

αni ch (kny) + βni sh (kny) cos (knx)

n=1

(18)

where ch() and sh() are hyperbolic cosine and sinus functions respectively. Note that the term α0i y + β0i appears since the eigenvalue of (14) can be zero [46].

In order to determine the parameters α0i , β0i , αni and βni , the remaining boundary conditions are used. From (16), the

following relations hold:

α01 = 0 and βn1 = 0

α4 = − µ0 and β4 = −α4 tanh (knH4) (19)

0

L

n

n

The next step consists in using the continuity condition

(17) to establish the link between the parameters of different

layers. The detailed derivation is given in Appendix, only the ﬁnal results are given here. The parameters α0i , β0i , αni and βni can be expressed by (24). Note that Fln,l+1 and Fln,l+1 −1 are the up-going and down-going transformation matrix, respectively:

Fl0,l+1 =

0 µl+1

µl

Hl 1

(20)

−1

Fl0,l+1 =

µl

0 µl+1

−µl

H 1 µl+1 l

(21)

Fln,l+1 =

ch (knHl) µµl+l 1 sh (knHl)

sh (knHl) µµl+l 1 ch (knHl)

(22)

5

l,l+1 −1

ch (knHl) − µµl sh (knHl)

Fn = −sh (knHl) µµl+ll+11ch (knHl) (23)

The parameter αn1 and αn4 can be derived from the continuity condition when i = j.

D. Magnetic Field Calculation

The ﬁrst integral can be performed on the conductor area where the current is non-zero. The second term corresponds to the average value of the potential vector on the boundary of the region, which is a constant. Neglecting this constant, the ﬁrst integral of (11) can be evaluated analytically [due to the term cos(knxs) in (25)] along x-axis and numerically along y-axis by discretizing the conductor into K thin elements (Fig.5). For element k, the y coordinate of the center point ysm = (yk + yk+1)/2 is chosen for the integration along y-axis.

The magnetic induction B can then be calculated by deriving Az, as given by (26). Note that the parameters α0i , β0i , αni and βni are only dependent to the coordinate of the source point. This property allows the separation of the mathematical treatment on the source conductor and the observation one. Therefore, the partial derivation on G (on the observation point) given in (26) can be performed directly on the expression whereas the integration on the source conductor is still evaluated numerically as done before.

∂Az

∂G (x, y)

Bx =

= −µ

J ds

∂y

Ω ∂y

(26)

∂Az

∂G (x, y)

By = − = µ

J ds

∂x

Ω ∂x

The ﬁnal purpose involves estimating the total energy in the cross-section via the integral (5). The Greens function has to be integrated both on the source conductor and the observation conductor. The integration on the source conductor is performed numerically while the integration on the observation conductor can be performed analytically according to the expression (18). However, three cases should be distinguished (Fig. 6):

1) The position of the source conductor element is higher than the observation conductor: In this case, the observation conductor is completely in layer i+1, therefore the integration is performed with the Greens function Gi+1 for layer i+1.

2) The position of the source conductor element is lower than the observation conductor: In this case, the observation conductor is completely in layer i, the integration is thereby calculated with the Greens function Gi for layer i.

3) The position of the source conductor element is included in the observation conductor: In this case, as the observation conductor crosses the two layers, the integration is performed by two separate parts. Both the Greens function Gi and Gi+1 should be integrated on the corresponding parts.

Fig. 5. Discretization along y-axis of the source conductor.

Fig. 6. Analytical integration on the observation conductor: Three cases.

Note that during the calculation of Az, a constant appears on the ﬁnal obtained results due to the Neumann type boundary conditions [47]. However, this constant will not affect the results of (5) since the integral of this constant over all the conductors will be null as the total ampere-turns is compensated in the component’s window.

E. Numerical Validations To validate the proposed method, the multi-layered Green’s

function is applied on two structures. The ﬁrst case (Fig. 7) does not correspond to real component but is computed to test the developped approach. The second one, is closer to a real planar component (Fig. 10(a)). For both components, numerical simulations, performed with ANSYS Maxwell 2D [48], are done to be compared to Green’s calculation. For each FEA simulation, total ampere-turns are compensated.

1) Conﬁguration test: As shown in Fig. 7, an FPC layer is implemented so that the structure is divided into three layers. The total ampere-turns of the component are compensated. To analyse the results, two test cuts are performed on this crosssection at y = 0.75 mm and y = 3.75 mm respectively. The obtained vector potential on these two test cuts are compared with Finite Elements simulations results, as shown in Fig. 8(a). As seen, a constant difference is observed between the calculated curves and the simulated ones. These results

6

Fig. 7. Test conﬁguration.

(a)

(b)

Fig. 8. Comparison between calculated results and simulated results. (a) Before removing the constant. (b) After removing the constant.

are foreseen due to the Neumann type boundary conditions. Removing this constant (Fig. 8(b)), a good agreement between these two curves can be found. The magnetic induction Bx and By can also be deduced, as illustrated in Fig. 9. One can note a good coherence between the calculated curves and simulated ones. After the numerical integration, the magnetic energy stored in the window is 2.29E − 5J/m while the calculated one is similar. In the previous analysis, it is supposed that the permeability is inﬁnite. Indeed, even though the practical permeability of ferrite material is in the order of 103µ0, the inﬁnite-permeability assumption still gives good approximation to the reality. FEA simulations have been performed to assess the inﬂuence of the magnetic core. Results are presented in Table I where the magnetic energy stored in the windows is computed with various permeability values. Comparing FEA results and the Green’s calculations, it can be seen that the proposed method can accurately determine the magnetic ﬁeld in the structures of interest.

2) Planar CM choke: This second test case is based on the design of a planar CM choke (Fig. 10(a)) made of 3F3 E32/I planar core [49]. Some FPC layer (height 0.8mm) is added between two windings. The total ampere-turns of the component are also compensated. The current in each conductor higher than the FPC layer is −1A whereas the current is 1A for the conductors lower than the FPC layer. The Fig 10(b) shows the magnetic ﬁeld inside the component’s window. From this simulation, it can be concluded that the ﬁeld is not tangential to the conductors. Therefore, the 1D

Dowell assumption is not directly applicable in such case. This hypothesis is conﬁrmed when analysing Bx and By (Fig. 11) along the horizontal test cut at H = 1.5mm. Simulated results are also compared to the computed ones in Fig. 11, where both simulated and calculated ﬁelds show good agreement.

After numerical integration, the calculated magnetic energy stored in the window is 2.43E − 5J/m, while the simulated value is 2.57E − 5J/m. The difference is equal only to 5%.

IV. APPLICATIONS

The previous method has been tested on two real components: A LLC transformer and a CM choke. Both devices are based on the use of ferrite planar cores [49], associated with FPC C350 [50] for the leakage layers.

A. Components’ description

For the LLC transformer (Fig. 12(a)), two 3F3 E38-cores are used while for the CM inductor (Fig. 13(a)), two 3C90 E43-cores are preferred.

The transformer is made of 4 copper layers of 35µm with 8 turns for the primary and 2 for the secondary. The CM inductor is a symmetrical component, made of 8 identical 70µm copper layers with 4 turns on each layer. For both components a leakage layer made of FPC is sandwiched between the two windings to increase the leakage inductance. This FPC layer is set to 0.2mm and 0.96mm, for the transformer and the CM inductor, respectively. The winding arrangements are described in Fig.12(b) and Fig.13(c).

0

α0i

=

0

β0i N−1 Fl0,l+1 −1

l=i

− µL0 0

i≤j and

i>j

1

Fl,l+1

αn1

αni

=

l=i−1

n

0

βni N−1

−1

α4

Fln,l+1

n

−α4 tanh (k H )

l=i

n

n4

j

−1

Fl0,l+1

l=3

j

Fl,l+1 −1

n

l=3

− µL0 β04

j

=

Fl0,l+1

l=1

0

− µj+1

β1 +

L

0

0

αn4 −αn4 tanh (knH4)

j

=

Fln,l+1

l=1

αn1 + 0

0 − 2µnjπ+1 cos (knxs)

i≤j i>j

(24)

(25)

7

TABLE I VARIATION OF MAGNETIC ENERGY WITH PERMEABILITY

µr

∞

10000 5000 1000

500

W smag(J/m) 2.29E-5 2.29E-5 2.29E-5 2.28E-5 2.27E-5

(a)

(a)

(b)

(c)

(d)

Fig. 9. Comparison between calculated results and simulated results for the test case. (a) Test cut 1: Bx. (b) Test cut 1: By. (c) Test cut 2: Bx. (d) Test cut 2: By.

(b)

Fig. 10. Planar CM choke cross section under simulation. (a) Window cross section. (b) Field inside the window.

B. Leakage inductance calculation

For the part inside the core, the previous Green’s computation is applied. For the winding part outside the core, the FPC layer is not implemented. Therefore, the PEEC-like formulations can be employed for calculating the magnetic ﬁeld as well as the magnetic energy per unit length. In fact, it can be shown that the PEEC-like formula (4) is the integral form of the free space Green’s function (Green’s theorem) over the rectangular region [−a/2, a/2] × [−b/2, b/2] [51]:

G (x, y; xs, ys) = − µ0 ln (x − xs)2 + (y − ys)2 (27) 2π

The Table II presents energy computed inside the core using Green’s functions Wic and the one calculated outside the core based on PEEC formulas Woc. The total leakage inductance is then calculated with (28) taking into account lengths of windings inside and outside the core (Fig. 14). For the LLC transformer, the leakage inductance is evaluated to 162nH while for the CM inductor, it is equal to 13.4µH.

2

Llk = I2 [2Wiclic + Woc(loc1 + loc2)]

(28)

Regarding computation time, duration depends on the num-

ber of conductors. For example, the LLC computation lasts 3.45s while the CM inductor 5.14s. As a comparison, FEA performed on the example of Fig. 10(a) lasts 15s. Such semi-

(a)

(b)

Fig. 11. Comparison between calculated results and simulated results. (a) Test cut: Bx. (b) Test cut: By.

analytical approach can then be interesting for component’s optimization and its leakage inductance tuning.

C. Leakage inductance measurement

Both components are then characterized with an impedance analyzer [52] and their leakage inductances are measured based on method proposed in [53]: The leakage impedance Zs (Fig. 15) is calculated using open-circuit and short-circuit impedance measurements (29). This method enables to cancel the effect on magnetizing inductance while calculating the leakage one (30).

Zs = Z0 (Zcc − rp) (29) (Z0 − rp) − rs

8

(a)

(b)

Fig. 12. LLC planar transformer (a) Component top view. (b) Cross section and dimensions.

Fig. 14. Typical winding length.

Fig. 15. Equivalent circuit for leakage measurement.

(a)

(b)

(c)

Fig. 13. Planar CM choke. (a) Component Top view. (b) Component Face view. (c) Cross section and dimensions

with Z0 the impedance measured from primary winding with secondary in open-circuit, Z0 the impedance measured from secondary winding with primary in open-circuit and Zcc the impedance measured from primary winding with secondary in short-circuit. rp (Fig.15) is deduced from Z0 in low frequency while rs is calculated with Z0 also in low frequency [53].

TABLE II LEAKAGE INDUCTANCE CALCULATION

Component Core type Mag. material

lic (m) loc1 + loc2 (m)

Wic (J/m) Woc (J/m)

Llk (H)

LLC E38 3F3 25.4E − 3 30E − 3 1.009E − 6 5.007E − 7 162E − 9

CM ind E43 3C90

27.9E − 3 88E − 3 6.691E − 5 3.367E − 5 13.4E − 6

Zs

Llkmeas

=

I m( 2πf

)

(30)

The measured values are plotted in Fig. 16, where they are compared to the computed ones. In this ﬁgure, for low frequencies, the leakage inductance is masked by resistances while in HF, parasitic capacitances cancel the magnetic effect. The leakage value is quite constant on the frequency range [1kHz; 1M Hz].

In both components coupling coefﬁcients k between windings are close to 1. Magnetizing inductances are evaluated to 431µH and 851µH for the LLC and the CM inductor, respectively. These values are widely stronger than the leakage ones. With such a difference, the total ampere-turn compensation hypothesis stays valid because magnetizing current can be neglected in case of short-circuit test.

D. Discussion

Regarding Fig. 16, it can be noted that computation for both cases gives good result with error lower than 15%. These differences can be justiﬁed for following reasons:

1) Errors related with dimensions: The dimensions reported in Fig. 12(b) and Fig. 13(c) have been measured directly on the prototypes and are not precises.

2) Simpliﬁcation of the geometry: The window’s description is not identical inside and outside the core. The part inside the magnetic core is well described but the one outside is simpliﬁed and the angles/corners are not really taken into account.

3) Sensitivity to the average length: The results are directly proportional to the energy density. They have to be multiplied by a lenght which is easy to determine inside the core (depth of the core) but which is more difﬁcult

9

to evaluate outside. A mean path is ﬁxed, based on window’s peak energy but this assumption is not precise. 4) Magnetic material disparities: As an example, the C350 material is given with a permeability of 9 ± 20% [50]. The same computation has been done again, taking into account this disparity. Calculation has been done with µr = 10.8(9 + 20%) and µr = 7.2(9 − 20%). The Table III summarizes the results and shows that this parameter can lead to signiﬁcant differences.

Without the leakage layers, leakage inductance can be evaluated, based on [39], to 128nH (Wic = 6.68E − 7J/m) and 11.8µH (Wic = 5.287E−5J/m) for the LLC transformer and the CM inductor, respectively. The leakage layers enable then to increase the leakage inductance and can be analytically calculated with a good precision with the developped method. However, this increase can be considered as quite low while regarding relative differences between leakage inductances with and without leakage layers. As a percentage, this rise is equal to 26% for the LLC and 13% for the CM inductor. These values can still be increased by acting on two main factors:

1) Increase of FPC thickness: In both examples, FPC layers are limited to 0.2mm and 0.96mm. Adding some layers, can enable to increase the leakage inductance value.

2) Addition of FPC outside the core: In this work, only FPC inside the core was considered. Adding leakage layers outside the core will also increase the leakage value. But the present method will not be able to calculate its value. Only 3D FEA could be performed to do this assessment.

These both solutions can also be discussed. Indeed, in most of power electronic converters, while power density is required, HF power transformers are made of interleaved layers to limit HF copper losses [54], [55]. Such interleaving reduces the leakage inductance. FPC can be added between each primary and secondary layers [56] but here also, the present method will be unusable. Some developpements have to be made on massively multi-layered windings as well as HF effects.

In connection with HF power transformers, problem of parallel windings should also be adressed. In planar transformers, parallel layers are a common practice to increase currents inside windings. The Green method can be applied with such conductor conﬁgurations. Indeed, only current values have to be settled, as in the example presented in Fig.7. In low frequency, the current repartition will be homogeneous and current will be uniformly distributed inside conductors. It is not true in HF, but the developped Green method is also limited and can not be applied in HF.

V. CONCLUSION

In this paper, the leakage inductance of planar magnetic components including leakage layer is studied via an analytical method based on multilayered Green’s function. Using this Green’s function, the Poisson equation for a rectangular region with Neumanns type boundary conditions can be solved to determine the magnetic ﬁeld in the cross-section of the studied component. Simulation and measurement results show that this

Fig. 16. Comparison between measurement and computation results.

method is very effective and accurate for planar component leakage inductance calculation when a leakage layer is inserted inside the component’s windings.

APPENDIX A DERIVATION OF EQUATION

Recall the general solution of Greens function in layer i (18) and the continuity condition (17). If i = j, δij = 0. In this case, the Dirac current source does not locate on the interface to analyze. The condition can be expressed by (31). Based on the equality of the coefﬁcients term by term, the following matrix systems can be derived:

αni+1 = Fi,i+1 αni

βni+1

n

βni

(n = 0, 1, 2...∞) (33)

Here the matrix Fin,i+1 is the upward transformation matrix that relates the parameters of layer i and layer i+1.

Its expression can be found in (20) and (22). The downward transformation matrix Fin,i+1 −1 can be deﬁned similarly as:

αni = Fi,i+1 −1 αni+1

βni

n

βni+1

(n = 0, 1, 2...∞) (34)

where Fin,i+1 −1 is given in (21) and (23). If i = j, δij = 1. In this case, the Dirac current source locates on the studied interface. The condition (17) becomes (32). Integrate the second term of this equation (32) on [0, L]:

L αj − L αj+1 = 1

(35)

µj 0 µj+1 0

Multiply the two sides of the second equation of (32) by cos(knx), and integrate it on [0, L]: